数字信号处理 清华大学出版社 chapter 1

eview for signals and systems
In signal and system analysis, the best approach is to represent the signal as a combination of some kind of most simplest signals which will pass through the system and produce a response. Combine the responses of all simplest signals, which is the system response of the original signal. Signal and system analysis is based on signal decomposition.
1 [n] 0 t 0 t0
1 0 1 2
n
A discrete-time signal x[n] can be expressed in terms of the unit impulse:
x[n]
k
x[k ] [n k ]

Review for signals and systems
c1 c 2
c 2 c1
c1 c 2

Ideal bandpass
Ideal bandstop
Review for signals and systems
Generation of Complex Signals In the nature world, most of signals are real-valued signals, called real signals. In some applications, it is desirable to develop a complex signal from a real signal having more desirable properties.
x(t )
Hilbert transformer
1 ˆ (t ) hHT (t ) x t
j
Its output response is
H HT ( j)

Its spectrum is
x( ) ˆ (t ) x( )hHT (t )d x d t
Review for signals and systems
Unit impulse function δ(t) is the simplest signal



(t )dt 1
t0
0
(t )
t
(t ) 0
A continuous-time signal x(t) can be expressed in terms of the unit impulse:
Review for signals and systems
There are four typical filters:
1
H ( j )
H ( j )
Ideal lowpass
H ( j )
1
c
c

1
c
c

Ideal highpass

1
H ( j )
c 2 c1
x(t ) x( ) ( t )d

Review for signals and systems
For an LTI system, if the input is δ(t), then the output signal is called the impulse response, denoted as h(t).
ˆ ( j) Y ( j) X ( j) jX 2 X p ( j)
Review for signals and systems
Physical meanings of the Hilbert transform For a real signal x(t), its Fourier transform X(jΩ) is not causal; By the use of the Hilbert transform, we can build an analytical signal which has a causal magnitude spectrum:
Review for signals and systems
Filtering and Filter One of the most widely used complex signal processing operations is filtering. Filtering is used to pass certain frequency components in a signal through the system without any distortion and to block other frequency components. The system implementing this operation is called a filter.
x[n] LTI system h[n] y[n]
y[n]
k
x[k ]h[n k ]

Review for signals and systems
Typical Signal Processing Operations Elementary Time-Domain Operations There are three most basic time-domain signal operations both for continuoustime and discrete-time signals. For continuous-time signals: 1. Scaling 2. Integration 3. Addition For discrete-time signals: 1. Scaling 2. Unit delay 3. Addition
Chapter 1
Review for Signals and Systems
Review for signals and systems
Any problem about signal analysis and processing may be thought of letting signals pass through systems. y(t) From x(t) and h(t), find x(t) h(t) y(t), Signal Processing From x(t) and y(t), find h(t), System Design From y(t) and h(t), find x(t), Signal Reconstruction
j H HT ( j) j 0 0
j
1 hHT (t ) t
H HT ( j)
j

Review for signals and systems
For a real signal x(t), its magnitude spectrum exhibits even symmetry, and its phase spectrum exhibits odd symmetry. Thus the spectrum X(jΩ) can be expressed as a sum of two parts:
X ( j) X p ( j) X n ( j)
X ( j )
X p ( j) X ( j) u()
X n ( j) X ( j) u()

X n ( j) X p ( j)
Review for signals and systems
If this real signal x(t) passes through a Hilbert transformer:
δ(t) LTI system h(t)
A continuous-time LTI system can be uniquely specified by its impulse response h(t).
x(t)
LTI system h(t)
y(t)
y(t ) x( )h( t )d
o
( A / 2)
Y ( j )
o
o
Review for signals and systems
Synchronous demodulation
y(t )
r (t )
A x (t ) 2
ˆ (t ) y(t ) x(t ) jx
ˆ (t ) is the Hilbert transform of x(t) where x
Review for signals and systems
Modulation and Demodulation DSB-Modulation x(t )
Review for signals and systems
Hilbert transformer
x(t )
Hilbert transformer
ˆ (t ) x
The impulse response of the Hilbert transformer is hHT(t) defined by Its frequency response:

1

j
ˆ ( j) X ( j) H ( j) jX ( j) jX ( j) X HT n p
Review for signals and systems